Skew-Hadamard matrices and the Smith normal form (Q1378886)

The authors show that a skew-Hadamard (s-H) matrix of order \(4m\) has the Smith normal form (Snf): \[ \text{diag}\Biggl\{1,\underbrace{2,\dots,2}_{2m- 1},\underbrace{2m,\dots,2m}_{2m- 1},4m\Biggr\}. \] The Snf of the incidence matrix of an s-H \((4m-1, 2m,m)\)-design and of the (complementary) s-H \((4m-1,2m-1,m-1)\)-design are: \[ \text{diag}\Biggl\{\underbrace{1, \dots,1}_{2m- 1}, \underbrace{m,\dots, m}_{2m- 1},2m\Biggr\} \] and \[ \text{diag}\Biggl\{\underbrace{1,\dots, 1}_{2m}, \underbrace{m,\dots, m}_{2m- 2},m(2m- 1)\Biggr\}, \] respectively.